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Additional info for Alkhathaami - Porosity Permeability Skin Factor- gPG

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We want to solve the equation ~(x) =b with bElRn given and ~* b on a Assume that 0 n ~-l(b) is ~inite and that the Jacobian matrix ~'(x) is nonsingular at these points. Then we can de~ine o. where I ~'(x) I is the determinant o~ the Jacobian matrix. Properties (i) d(I,O,b)=l o~ the degree. i~ bE 0, d(I,O,b) =0 i~ b¢O; (ii) i~ ~*b in 0 then d(~,O,b) =0; (iii) (Additivity) i~ 0=0 1 u O2 , b¢~(a Ou 01 n0 2 ) then d(~,O,b) =d(~,Ol,b) +d(~,02,b); (iv) (Continuity) d is continuous in ~,O and b. Note that since d(~,O,b) =d(~-b,O,O) it su~~icies to veri~ the continuity with respect to ~ and O.

1) has no solution on Ox - Br (A) and then ,1 ° ° M. G. CRANDALL AND P. H. RABINOWITZ 32 u fig. 19 for A>P. 2) holds also for A < P . ),Ox,O) = canst. for I A-P I < l). Now suppose p-l)< p< ii

_ (vi) Define fX = I-XL where L is an nxn matrix. fX vanishes at zero and when X is a characteristic value of L. Then, if Br in the ball of radius r and center 0, d(fX,Br,O) is well defined as long as A is not a characteristic value. (rx) =0). Furthermore d(fX, Br,O) changes by a factor (_l)m as A crosses a characteristic value of mUltiplicity m. ,Br,O) = sign det(I-AL). The degree may be extended to functions on a Banach space E. i,E) with nbounded and open subset of E and T is compact. If bEE and b f1 4>( an) we may define d(cI>,n,b) by approximating 4> with mappings over finite-dimension-.